Compressed Secret Key Agreement: Maximizing Multivariate Mutual Information Per Bit

The multiterminal secret key agreement problem by public discussion is formulated with an additional source compression step where, prior to the public discussion phase, users independently compress their private sources to filter out strongly correlated components for generating a common secret key. The objective is to maximize the achievable key rate as a function of the joint entropy of the compressed sources. Since the maximum achievable key rate captures the total amount of information mutual to the compressed sources, an optimal compression scheme essentially maximizes the multivariate mutual information per bit of randomness of the private sources, and can therefore be viewed more generally as a dimension reduction technique. Single-letter lower and upper bounds on the maximum achievable key rate are derived for the general source model, and an explicit polynomial-time computable formula is obtained for the pairwise independent network model. In particular, the converse results and the upper bounds are obtained from those of the related secret key agreement problem with rate-limited discussion. A precise duality is shown for the two-user case with one-way discussion, and such duality is extended to obtain the desired converse results in the multi-user case. In addition to posing new challenges in information processing and dimension reduction, the compressed secret key agreement problem helps shed new light on resolving the difficult problem of secret key agreement with rate-limited discussion, by offering a more structured achieving scheme and some simpler conjectures to prove

Algorithms for weighted multidimensional search and perfect phylogeny

This dissertation is a collection of papers from two independent areas: convex optimization problems in R[superscript]d and the construction of evolutionary trees;The paper on convex optimization problems in R[superscript]d gives improved algorithms for solving the Lagrangian duals of problems that have both of the following properties. First, in absence of the bad constraints, the problems can be solved in strongly polynomial time by combinatorial algorithms. Second, the number of bad constraints is fixed. As part of our solution to these problems, we extend Cole\u27s circuit simulation approach and develop a weighted version of Megiddo\u27s multidimensional search technique;The papers on evolutionary tree construction deal with the perfect phylogeny problem, where species are specified by a set of characters and each character can occur in a species in one of a fixed number of states. This problem is known to be NP-complete. The dissertation contains the following results on the perfect phylogeny problem: (1) A linear time algorithm when all the characters have two states. (2) A polynomial time algorithm when the number of character states is fixed. (3) A polynomial time algorithm when the number of characters is fixed

Two Studies in Representation of Signals

The thesis consists of two parts. In the first part deals with a multi-scale approach to vector quantization. An algorithm, dubbed reconstruction trees, is proposed and analyzed. Here the goal is parsimonious reconstruction of unsupervised data; the algorithm leverages a family of given partitions, to quickly explore the data in a coarse-to-fine multi-scale fashion. The main technical contribution is an analysis of the expected distortion achieved by the proposed algorithm, when the data are assumed to be sampled from a fixed unknown probability measure. Both asymptotic and finite sample results are provided, under suitable regularity assumptions on the probability measure. Special attention is devoted to the case in which the probability measure is supported on a smooth sub-manifold of the ambient space, and is absolutely continuous with respect to the Riemannian measure of it; in this case asymptotic optimal quantization is well understood and a benchmark for understanding the results is offered. The second part of the thesis deals with a novel approach to Graph Signal Processing which is based on Matroid Theory. Graph Signal Processing is the study of complex functions of the vertex set of a graph, based on the combinatorial Graph Laplacian operator of the underlying graph. This naturally gives raise to a linear operator, that to many regards resembles a Fourier transform, mirroring the graph domain into a frequency domain. On the one hand this structure asymptotically tends to mimic analysis on locally compact groups or manifolds, but on the other hand its discrete nature triggers a whole new scenario of algebraic phenomena. Hints towards making sense of this scenario are objects that already embody a discrete nature in continuous setting, such as measures with discrete support in time and frequency, also called Dirac combs. While these measures are key towards formulating sampling theorems and constructing wavelet frames in time-frequency Analysis, in the graph-frequency setting these boil down to distinguished combinatorial objects, the so called Circuits of a matroid, corresponding to the Fourier transform operator. In a particularly symmetric case, corresponding to Cayley graphs of finite abelian groups, the Dirac combs are proven to completely describe the so called lattice of cyclic flats, exhibiting the property of being atomistic, among other properties. This is a strikingly concise description of the matroid, that opens many questions concerning how this highly regular structure relaxes into more general instances. Lastly, a related problem concerning the combinatorial interplay between Fourier operator and its Spectrum is described, provided with some ideas towards its future development

Discrete Geometry (hybrid meeting)

A number of important recent developments in various branches of discrete geometry were presented at the workshop, which took place in hybrid format due to a pandemic situation. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics, algebraic geometry or functional analysis. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

Parametric shortest-path algorithms via tropical geometry

We study parameterized versions of classical algorithms for computing shortest-path trees. This is most easily expressed in terms of tropical geometry. Applications include shortest paths in traffic networks with variable link travel times.Comment: 24 pages and 8 figure

A New Dynamic Programming Approach for Spanning Trees with Chain Constraints and Beyond

Short spanning trees subject to additional constraints are important building blocks in various approximation algorithms. Especially in the context of the Traveling Salesman Problem (TSP), new techniques for finding spanning trees with well-defined properties have been crucial in recent progress. We consider the problem of finding a spanning tree subject to constraints on the edges in cuts forming a laminar family of small width. Our main contribution is a new dynamic programming approach where the value of a table entry does not only depend on the values of previous table entries, as it is usually the case, but also on a specific representative solution saved together with each table entry. This allows for handling a broad range of constraint types. In combination with other techniques -- including negatively correlated rounding and a polyhedral approach that, in the problems we consider, allows for avoiding potential losses in the objective through the randomized rounding -- we obtain several new results. We first present a quasi-polynomial time algorithm for the Minimum Chain-Constrained Spanning Tree Problem with an essentially optimal guarantee. More precisely, each chain constraint is violated by a factor of at most $1+\varepsilon$, and the cost is no larger than that of an optimal solution not violating any chain constraint. The best previous procedure is a bicriteria approximation violating each chain constraint by up to a constant factor and losing another factor in the objective. Moreover, our approach can naturally handle lower bounds on the chain constraints, and it can be extended to constraints on cuts forming a laminar family of constant width. Furthermore, we show how our approach can also handle parity constraints (or, more precisely, a proxy thereof) as used in the context of (Path) TSP and one of its generalizations, and discuss implications in this context.Comment: A short version of this work appeared in the proceedings of the 30th annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2019